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Local Scott compactification
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
(English)Article in journal (Refereed) Submitted
Abstract [en]

We show how to embed certain formal topologies in locally Scott formal topologies. We call this process local Scott compactification. Examples include localic completions of metric spaces. We also prove a lifting result for morphisms. The local Scott compactification of a space corresponds to a domain representation of the space and in this way yields a space containing partial elements. In the case of the real numbers we obtain the space of partial reals, consisting of intervals where the endpoints are lower and upper reals respectively. We show that partial reals that define compact overt subspaces of the real numbers correspond precisely to intervals with real endpoints. The lifting of morphisms give sharp extensions of continuous real valued functions in the sense of interval analysis. These results are also generalized to normed vector spaces.

Keyword [en]
Formal topologies, Continuous dcpos, Locally compact metric spaces
National Category
Algebra and Logic
Research subject
Mathematical Logic
Identifiers
URN: urn:nbn:se:uu:diva-152064OAI: oai:DiVA.org:uu-152064DiVA: diva2:412411
Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14Bibliographically approved
In thesis
1. Contributions to Pointfree Topology and Apartness Spaces
Open this publication in new window or tab >>Contributions to Pointfree Topology and Apartness Spaces
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. In Paper I we focus on the notion of a domain representation of a formal space as a way to introduce generalized points of the represented space, whereas we in Paper II give a constructive and point-free treatment of the domain theoretic approach to differential calculus. The last two papers are of a slightly different nature but still concern constructive topology. In paper III we consider a measure theoretic covering theorem from various constructive angles in both point-set and point-free topology. We prove a point-free version of the theorem. In Paper IV we deal with issues of impredicativity in the theory of apartness spaces. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2011. 40 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 71
Keyword
Constructive mathematics, General topology, Pointfree topology, Domain theory, Interval analysis, Apartness spaces
National Category
Algebra and Logic
Research subject
Mathematical Logic
Identifiers
urn:nbn:se:uu:diva-152068 (URN)978-91-506-2219-5 (ISBN)
Public defence
2011-06-08, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2011-05-17 Created: 2011-04-23 Last updated: 2011-06-14Bibliographically approved

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